I'm not sure what you mean by "...do not have knowledge of the structure of"

I'm not sure what you mean by "...do not have knowledge of the structure of" — Moliere

Yes, me too.

I mean we know the premises, but don't know the format of the argument. Such as whether it is modus tollens or ponens (if you couldn't just guess between those two based on the premises and conclusion).

It isn't so much a reference to lexical semantics and whatnot, but that might matter too.

Can anyone give me an example of an argument that we know is deductively valid and has correct premises but do not have knowledge of the structure of? Is there a sub-branch of philosophy that deals with such things? I can't find anything on it anywhere. — ToothyMaw

What makes you think there is such a thing?

Virtually any mathematical conjecture would be of this type. When I compose a possible theorem I'm not certain about the argument I will ultimately use, although I am almost sure it is correct - but not absolutely. This is true of most mathematicians. Fermat's Last Theorem was assumed true long before the proof was established. But no one was absolutely certain.

I feel like it exists.

Is it possible to break down a deductively valid mathematical theorem into its constituent parts without knowing the actual argument that is going to be used to prove it? In other words: what if the reasoning cannot be observed at all but we know it's (correct) constituent parts when synthesized add up to a deductively valid argument?

Well, I might think, I'll use mathematical induction and see where that goes. And it might work, or not. I can essentially do a simple proof in BASIC and see all the constituent parts, and I might have thought of the various parts before designing my program. But to know for certain the following 10 parts will be used before I begin experimenting with them is a stretch. And a rather foolish and non-productive expenditure of energy.

Thanks. You basically answered my question.

If there was a way to determine if it is possible at all to observe or determine the proof behind a certain unproven but assumedly correct principle or theorem, would that be valuable?

Well, yes. If God spoke to me saying, THIS THEOREM IS TRUE, then that would remove a nagging uncertainty in the proof process. But it might end up making little difference in the future effort. On the other hand, if He said, DON"T WASTE YOUR TIME, I wouldn't. :cool:

Interesting. What if it applied conversely and allowed us to determine that the proof behind a principle or theorem assumed to be true couldn't be observed or determined?

What if it applied conversely and allowed us to determine that the proof behind a principle or theorem assumed to be true couldn't be observed or determined? — ToothyMaw

If a theorem is "true", then it is so subject to foundational axioms. Changing the axioms could change the designation of "true". "True" means derivable in this way. "True" isn't out there all by itself.

I get that, I'm talking about a way to determine if said theorem can be understood by observing the proof. I do not deny the fact that the proof would be a function of certain foundational axioms if it is true, thus meaning the proof is derivable given those axioms. What I'm talking about is like looking in through a window that may or may not be there on something that we know is there (the proof).

I'm no expert on proofs, I hope I got that right.

The statement of a theorem is a way of understanding what has been proved. In modern abstract mathematics, however, one would need advanced knowledge to understand even that statement.

It is possible for the researcher to argue step by step, building a logical edifice, without stating the theorem. Then stating the formal structure of the theorem by say, "Thus we find that . . . ". I just did that in a math note, in fact:

"
A Brief Analysis . . ." (page 2)

a sub-branch of philosophy — ToothyMaw

In mathematics (and often crossing into philosophy) there is the subject of mathematical logic in which the notions of truth and provability are given rigorous explication.

First we need to be very clear in our terminology:

Formally, an argument is merely an ordered pair <G P> where G is a set of statements and P is a statement. G is the set of premises and P is the conclusion.

An argument is valid if and only if there is no model in which every member of G is true but P is false. So 'validity of an argument' is a semantical notion.

This formal sense of 'argument' is different from the informal sense of an argument being a presentation of reasoning in order to demonstrate a proposition.

On the other hand, formally, a proof is a sequence of statements such that every statement in the sequence is either an axiom (or an axiom or premise) or follows by the deduction rules from previous statements in the sequence. So this 'proof' is a syntactical notion.

However, informally, mathematicians commonly use the rubric 'proof' to mean something along the lines of 'a correct and convincing argument', in the informal sense of 'argument'. Arguably, in almost any case, such an informal proof is one that can be translated into a formal proof.

we know the premises, but don't know the format of the argument — ToothyMaw

Suppose we do know the "format", i.e. we are given the premises and the formal sequence of steps toward the conclusion, then there is an algorithm that will determine whether the steps are correct so that we know that indeed the sequence is a proof of the conclusion from the premises.

However, even if we are not given the sequence of steps, in some cases we can still demonstrate that the conclusion is entailed by the premises, either by filling in steps ourselves or by making a semantical argument that the premises entail the conclusion.

In some other case though, we might not happen to know how to see whether the premises entail the conclusion, and there can be no general algorithm that determines whether any given statement is entailed by a given set of premises (this is undecidability, which stems from Godel's incompleteness, et. al).

In any case, notice that whether the premises are true or not is a separate from the questions of validity and proof. Validity of an argument is only that any model in which the premises are true is a model in which the conclusion is true.

And a conclusion may be proven from a set of premises, irrespective of whether or not the axioms or premises are true in a given model; this is the "relative" nature of proof - our mechanics of proof only provide validity and don't ensure truth.

Virtually any mathematical conjecture would be of this type. When I compose a possible theorem I'm not certain about the argument I will ultimately use, although I am almost sure it is correct - but not absolutely. This is true of most mathematicians. Fermat's Last Theorem was assumed true long before the proof was established. But no one was absolutely certain. — jgill

What's the "truth rate" of mathematical (or other) conjectures, herein defined as the percentage of them that have been found to be true?

If it's high enough then something mighty interesting is going on:

1. Truthiness

2. Verisimilitude

3. Intuition

Truths probably have some quality other than a formal, logical, proof that's a dead giveaway - we instantly recognize them, based on that quality whatever that is.

We could be looking at something revolutionary here; we wouldn't need to prove truths the old way, via argumentation. It just feels true, don't it?

Jean-Baptiste Biot, who assisted Laplace in revising it for the press, says that Laplace himself was frequently unable to recover the details in the chain of reasoning, and, if satisfied that the conclusions were correct, he was content to insert the constantly recurring formula, "Il est aisé à voir que ... " ("It is easy to see that ..."). — Wikipedia

Google or visit wikipedia for more!

I think I want a simpler example, and I believe I'm tracking now.

The classic argument --

All men are mortal
Socrates is a man
Therefore, Socrates is mortal

really does seem to make sense, but technically it's not deductively valid. "Socrates" is not well defined in Aristotle's system of logic, because it is a unique thing rather than the member of a set -- and Aristotle's logic deals with the relationships between sets (at least, this is a certain way of parsing it. It deals with the deductive relationship between statements of the "all" or "some" kind)

So, I'd say, here we have an example of an argument that makes sense, but which you could argue into different formalization of logic -- so it's not like we know the "correct" relationships to choose, up front.

Formally, an argument is merely an ordered pair <G P> where G is a set of statements and P is a statement. G is the set of premises and P is the conclusion.

An argument is valid if and only if there is no model in which every member of G is true but P is false. So 'validity of an argument' is a semantical notion. — TonesInDeepFreeze

What could be said about an argument, A, whose premises include the entire set of the correct premises of sound argument B, and has the same conclusion as B, but the conclusion is unsound for A, potentially because of added steps or premises? Doesn't that give a model in which every member of G could be true but the conclusion, P, be false?

Or what if there is some sort of recursive step in a valid model, an instance in which correct premises are applied to correct premises in such a way that the conclusion P of argument G becomes false even though all the premises stay true? Would that not be a model that would defy the formal definition?

Potentially good example, thanks. But:

I disagree on the argument not making sense deductively (disregarding the hidden premise of the argument)*. While Aristotle might be the only person who is the unique collection of all the traits he possesses, he still belongs to a number of sets according to those traits, unless I'm just in ignorance of the way Socrates is treated by Aristotle. He is, for instance, a man, but I don't think that needs to be assumed up front before the argument is made.

*There needs to be a premise saying that not all men are Socrates, or it doesn't make sense.

However, I think I see what you are saying: Socrates is unique insofar as we cannot treat him as just a man; he is not assumed to be a subset of anything, so, therefore, because of the second premise, we are no longer dealing with sets, or "all" and "some" statements, which goes against the way Aristotle did logic.

So, while this argument might make sense deductively, it doesn't make sense as part of a larger schema.

added steps — ToothyMaw

For this discussion, in order to be as clear as possible, I suggest sticking with my technical distinction between an argument and a proof, even though in ordinary discussions we don't make that technical distinction.

An argument is an ordered pair <G P> where G is a set of sentences and P is a sentence. G is the set of premises and P is the conclusion.

An argument is valid if and only if there is no model in which every member of G is true but P is false.

An argument is sound (per a given model) if and only if the argument is valid and every premise is true (in said model).

A proof is a sequence of formulas such that every line in the sequence is either an axiom or a premise or follows by the deduction rules from previous lines in the sequence. (Note that, contrary to ordinary usage, there is no such thing as an "incorrect" proof. Something is either a proof or not. If a sequence has a line that is not an axiom, premise or follows by the deduction rules from previous lines in the sequence, then that sequence is not an "incorrect" proof but rather it simply is not a proof.)*

* Note that I am simplifying by not mentioning that there are other proof forms such as sequents (not to be confused with sequences), trees, natural deduction proofs (which can be rendered as sequences of lines that each have an index number, a formula, and the set of index numbers that are the undischarged assumptions for that line), and probably others.

/

an argument, A, whose premises include the entire set of the correct premises of sound argument B — ToothyMaw

I don't know what you mean by "correct" premise. So I'll take the above as just saying:

A and B are arguments with the same set of premises.

and has the same conclusion as B — ToothyMaw

Then A and B are the same argument, since they have the same premises as each other and the same conclusion as each other.

but the conclusion is unsound for A — ToothyMaw

I wouldn't use the terminology "the conclusion is unsound for the argument". Maybe you mean that the conclusion is not entailed by the premises, i.e. the argument is not valid.

because of added steps or premises — ToothyMaw

Steps pertain to proofs not to arguments. Proofs have steps; arguments don't have steps.

So let's look at proofs. If you have a proof and add steps that are not needed, it's still a proof. There is no requirement that every step in a proof must be a needed step.

Now, every proof is relative to its axioms and premises. So let H be the set of axioms and premises that are mentioned in the proof and let T be the last line in the proof, then we say the proof is a proof of T from H.

Now, back to arguments, a very very important thing to keep in mind about classical logic is that it is monotonic. This means that if an argument <G P> is valid than any argument <H P> is valid too where G is a subset of H. In other words, adding premises to a valid argument still results in a valid argument. That holds without exception in classical logic.*

* As an aside, this should be recognized to hold even informally in such cases where there is a valid argument but an ad hominem is gratuitously added . For example, suppose the argument is:

Premises = {"All fish are creatures", "All trout are fish", "My debate opponent is stupid"}.
Conclusion = "All trout are creatures".

That is a valid argument, despite that there is an ad hominem among the premises. That's because the logic is monotonic. The argument without the ad hominem is valid, so adding another premise doesn't result in invalidity.

Doesn't that give a model in which every member of G could be true but the conclusion, P, be false? — ToothyMaw

No.

Also, better not to say "could be true", which complicates with subjunctive modality. Just say "is true".

recursive step in a valid model — ToothyMaw

I don't know what you mean by a "step in a model" and even more I don't know what you mean by a "recursive step in a model".

an instance in which correct premises are applied to correct premises — ToothyMaw

Again, I don't know what you mean by "correct premise".

Per a given model, a sentences is either true or false (and not both true and false).

So sentences come in these varieties:

logically true (i.e. valid*, i.e. true in every model)

not logically true (i.e. not valid, i.e. false in at least one model).

logically false (i.e. false in very model)

And a sentence that is not logically true can be either logically false (false in every model) or contingent (true in at least one model but also false in at least one model).

* Notice that the word 'valid' has two different contexts: (1) A valid argument, as defined earlier, and (2) a valid sentence, as defined just above . (Also, there is the notion of valid formulas, but that requires explaining the technical difference between a formula and a sentence, and the difference between satisfaction of a formula per a model and an assignment for the variables and truth of a sentence per a model.)

in such a way that the conclusion P of argument G becomes false even though all the premises stay true? — ToothyMaw

Sentences don't change truth value per arguments. Rather, contingent sentences have different truth values per models.

Would that not be a model that would defy the formal definition? — ToothyMaw

I don't know in what sense something could be a model though "defying the formal definition".

Models, arguments, and proofs are different things, though related.

I'm obviously in too deep here, I'll need to check out some actual mathematical logic books or something. Could you offer a starting point maybe? I have the Book of Proof.

Let's say for context that we are interested in a particular model M at some point in discussion. So temporarily we'll take 'true' and 'false' to stand for 'true in M' and 'false in M'.

Note:

An argument can be valid even if the conclusion is false. In this case, at least one of the premises would also have to be false, in which case it is not a sound argument though it is a valid argument.

An argument can be valid even if one or more of the premises are false. In this case it is not a sound argument though it is a valid argument.

An argument is not valid only when there is at least one model in which all the premises are true and the conclusion is false.

Absolutely I can recommend the very best textbooks I have found after looking at and reading many of them:

First. Learn symbolic logic. How formulas and formal sentences are made. How many English expressions can be translated into formulas. The basic notion of an interpretation (basically a model). How formal proofs are made in first order predicate logic using a natural deduction system. The book to get (it will empower you with the basic tools for this subject and for critical thinking in general):

Logic: Techniques Of Formal Reasoning - Kalish, Montague and Mar

Next. For the very best explanation of the method of definitions in logic and mathematics, study chapter 8 of:

Introduction To Logic - Suppes

Next. Learn axiomatic set theory. Mathematical logic is itself couched in the terminology and principles of set theory, so learning set theory before mathematical logic is well advised. The book to get:

Elements Of Set Theory - Enderton.

Next. For mathematical logic, the book to get:

A Mathematical Introduction To Logic - Enderton

Next, as a supplement, for possibly the most authoritative overview of the subject of formal logic, read the Introduction chapter of:

Introduction To Mathematical Logic - Church

Thanks, man. Do I really have to read all the stuff before Enderton? I want to get to the meat of it right away, but if I really have to...

You can study Enderton's mathematical logic book first. However, the book is intended for upper division students who are already studying abstract mathematics such as analysis, abstract algebra, maybe topology. It's not that the material in those subjects is required, but that a level of mathematical sophistication and reasoning is obtained through study of that kind of mathematics.

On the other hand, I didn't study mathematics such as analysis, abstract algebra, or topology prior to studying the book. But I did study the sequence of books I recommended to you. And it worked perfectly for me. I even think that the sequence I recommend is better for leading up to Enderton's book than just studying abstract mathematics.

But if you just start with the Enderton logic book, you'll probably grasp some, maybe a lot, of it. But without the books I recommend first, I bet you'll still be confused about a lot of the Enderton logic book. You surely won't get some of the concepts firmly. Most importantly, I don't think you'll really get what he's driving at without first really knowing how to work in symbolic logic.

When you study the Kalish-Montague-Mar book first, you really understand working in symbolic logic, and that will raise questions in your mind about looking at symbolic logic from above, about not just what it is involved in working in symbolic logic but about questions about certain properties of the logic system itself, i.e. not just the logic but the meta-logic. I.e. learn the logic first in Kalish-Montague-Mar then in Enderton learn the meta-theorems about that logic. The set theory book is also before the Enderton logic book, because with set theory you'll really see that the Enderton logic book is doing things rigorously from the set theoretic axioms.

On the other hand, a lot of people are a lot smarter than I am, and probably they could learn the Enderton book quite well without first reading those other books or first having having studied some upper division math.

For example, even at page 9, just a few pages in, he writes (Zorns' lemma):


"Say that a collection of sets C is a chain iff for any elements x and y of C, either x is a subset of y or y is a subset of x.

Let A be a nonempty set such that for any chain C subset of A, the set union-of_C is in A. Then there is some element m in A which is maximal in the sense that it is not a subset of any other element of A."


Granted, that doesn't come into play very much in the book (because it's not needed for countable languages), but you see that there's a level of mathematical sophistication taken for granted from the start. On the other, having first studied his set theory book, you'd know Zorn's lemma, its proof, its relationship with the axiom of choice and the well ordering theorem, and its importance generally.

I'm curious what you think of Schaums Outline of Logic. I found math outlines that worked well in several courses I taught. Would you recommend this for students such as ToothyMaw? I've never looked at a copy.